The Second Generation Fukui Project and a New Application of the Asymptotic Linearity Theorem †

the Asymptotic Linearity Theorem

Shigeru Arimoto*

The Asymptotic Linearity Theorem (ALT), which proves the Fukui conjecture in a broader context, plays a significant role in the repeat space theory (RST) - the central unifying theory in the first and the second generation Fukui Project. The present paper provides a new application of the ALT. This application is important for a new development of the RST towards the solution of Fukui’s DNA problem. The present paper also reviews the notions of the normed repeat space Xr(q, d, p) and its super space XB(q, d, p), which are special Banach algebras fundamental in the second generation Fukui project.

KEYWORDS: the Fukui conjecture, repeat space theory (RST), Asymptotic Linearity Theorem (ALT), Banach algebras

1. INTRODUCTION

In his later years, Kenichi Fukui (1918 - 1998, Nobel Prize 1981) presented several conjectures concerning the additivity problems of molecules having many identical moieties. Among them is the following which has been playing a significant role in the development of the repeat space theory (RST) (cf. [1-18]), which is the central unifying theory in the first (cf. [1,2]) and the second (cf.

[2,3]) generation Fukui Project:

The Fukui Conjecture. Let {MN} be a fixed element of the repeat space with block-size q, and let I be a fixed closed interval on the real line such that I contains all the eigenvalues of MNfor all positive integers N. Let _1/2: I 

denote the function defined by _1/2(t) = |t|^1/2. Then, there exist real numbers and  such that

Tr_1/2(MN) =  N+ + o(1) (1.1) as N .

Fukui's DNA problem, which is closely related to the Fukui conjecture above, is a long-range target of the first and second generation Fukui Project, whose underlying motive has been to cultivate a new interdisciplinary region between chemistry and mathematics, especially for tackling what we call globally-pertaining-type problems, or, for short, g-type problems [2]; these constitute physicochemical problems for whose solutions global mathematical contextualization is essential. "Can the conductivity and

Received August 31, 2010

*Division of General Education and Research

†This article is dedicated to the memory of the late Professors Kenichi Fukui and Haruo Shingu.

other properties of a single-walled carbon nanotube be analyzed in the setting of a -algebra equipped with a complete metric?" This metric problem is fundamental to proceed towards the solution of Fukui's DNA problem. In recent publication [3] by the present author, this metric problem was affirmatively solved and the new notion of normed repeat space X_r(q, d, p) was established. The normed repeat space X_r(q, d, p) is an intermediate theoretical device to shift from periodic polymers to aperiodic polymers like DNA and RNA in the Fukui Project.

The space X_r(q, d, p) is a Banach algebra for all 1 ≤ p ≤

and X_r(q, d, p) forms a C*-algebra for p = 2. Here, polymer moiety size number q and dimension number d are arbitrarily given positive integers. The generalized repeat space X_r(q, d) is contained in the normed repeat space X_r(q, d, p), which in turn is contained in one of its super spaces X_B(q, d, p) so that aperiodic polymers can be represented and investigated within this super space X_B(q, d, p).

The normed repeat space X_r(q, d, p) and its super space X_B(q, d, p) are fundamental in the second generation Fukui project.

Let q be any positive integer, the repeat space with block-size q, given in the Fukui conjecture stated above, is denoted by Xr(q). Let X(q) denote the set of all matrix sequences whose N-th term is an arbitrary qN  qN real symmetric matrix. Then, one can easily verify that the repeat space Xr(q) with block-size q is expressed by Xr(q) = X(q)  X_r(q, 1). Thus, we have the following relations between the repeat space Xr(q) in the Fukui conjecture, generalized repeat space X_r(q, 1), normed repeat space X_r(q, 1, p) and its superspace X_B(q, 1, p) (cf. the appendix for the definitions of the latter three spaces):

Xr(q) X_r(q, 1) X_r(q, 1, p) := closure of X_r(q, 1) 

X_B(q, 1, p). (1.2)

The Asymptotic Linearity Theorem (ALT) plays a significant role in the repeat space theory. This theorem, which was proved by the present author for the first time (cf.

[12] and references therein), implies the validity of the Fukui conjecture; combined with its associated theorems, it solves a variety of molecular network problems in a unifying manner (cf. [9,12] and references therein).

We retain the notation of [12]. (The reader is asked to briefly review [12] for the definition of symbols.) The present paper provides a new application of the ALT. The following theorem 1, from which fundamental theorem I in [1] easily follows, is important in a new development of the repeat space theory especially towards the solution of Fukui’s DNA problem. The goal of the present paper is to give an affirmative answer to the following problem.

Problem 1. Is it possible to derive theorem 1 by using the ALT?

Theorem 1. Let a, b  with a < b, let x(N, r) :=

a + (b - a)r/N, and let f  AC[a, b]. Then, we have

1 N



a

b f()d)N + (1/2)(f(b) - f(a))

+ o(1) (1.3)

as N .

In section 2, we prepare some tools for the affirmative solution to the above problem. The solution is given in section 3 by deriving, from the ALT, the following theorem 2, from which theorem 1 easily follows.

Theorem 2. Let a, b  with a < b, let x(N, r) :=

a + (b - a)r/N, and let g let f  AC[a, b]. Then, there exist real numbers (f) and (f) such that

1 N



2. PREPARATION FOR A SOLUTION OF PROBLEM 1

Throughout, let ⁺, 0+, , , and , denote respectively the set of all positive integers, nonnegative integers, integers, real numbers, and complex numbers. Let us first recall the symbols we need in this section: Let a, b  with a < b, let I = [a, b]. The symbol C(I) denotes the set of all real-valued continuous functions on I. A function  : I  is said to be absolutely continuous on I if, given any  > 0, there

exists a > 0 such that for every finite system of pairwise disjoint subintervals (a1, b1), (a2, b2), ..., (an, bn)  [a, b],

k1



implies

k1



The symbol AC(I) denotes the set of all real-valued absolutely continuous functions on I.If f is a real-valued function on I and if S is a subset of I, then f| S denotes the function f restricted to S. In this article, the symbols C(I) and AC(I) used in [1,12] are often represented by C[a, b] and AC[a, b] respectively. The following propositions 1 and 2 are fundamental in the present article:

Proposition 1. Let a, b, c  with a < c < b. The following statements are true:

(i) If f  C[a, b], f| [a, c] AC[a, c], f| [c, b] AC[c, b], then f  AC[a, b].

(ii) Suppose that f  C[a, b] is a monotone non-decreasing function. Let h > 0, let dh: [a, b – h]  denote the function defined by dh(x) = f(x + h) – f (x). If dh is a monotone non-decreasing function for all h > 0, then f  AC[a, b]. If dh is a monotone non-increasing function for all h > 0, then f

 AC[a, b].

(iii) If f  AC[a, b] is a monotone non-decreasing function and if g  AC[f(a), f(b)], then g  f  AC[a, b].

Proof. The conclusions easily follow from the definition of

absolutely continuous functions. //

Proposition 2. Let g: [0, 1]  [0, 1] be a continuous monotone increasing function defined by

g(x) = sin²(x/2). (2.1) Then, the following statements are true:

(i) g^-1 AC[0, 1]. (2.2)

(ii) For any f  AC[0, 1] there exist real numbers (f) and

(f) such that

1 N



Proof. (i) One easily verifies this by proposition 1(i-ii). (One can also easily verify (i) by 1(i) and the well-known fact that if f  C[a, b] is a convex function then, f  AC[a, b].) (ii) Recall {KN} Xr(1) given by (2.2) in [9], and define {MN}  Xr(1)  XHr(1, 1) by

MN = (1/4)KN. (2.4)

Let f AC[0, 1]. Notice that the jth eigenvalue _j(MN) of MN, arranged in the increasing order, is given by

_j(MN) = sin²((j – 1)/(2N)), (2.5) and that

Tr f (MN) =

1 N





(2.6)

By the Asymptotic Linearity Theorem (Practical ALT, Xr(q)-version) reproduced below (cf. [11,12] and references therein for details), the conclusion directly follows. //

Theorem PALT (Practical ALT, Xr(q)-version). Let {MN}  Xr(q) be a fixed repeat sequence, let I be a fixed closed interval compatible with {MN}. Then, for any   AC(I), there exist (), ()  such that

TrMN) = ()N+ () + o(1) (2.7) as N.

Recall (1.2) and cf. (A.14) in the appendix for the definition of Xr(q); a closed interval I is called compatible with {MN}  Xr(q) if all the eigenvalues of MN are contained in I for all N ⁺ (cf. [12] and references therein for details).

In proof (ii) of proposition 2 above, one can also apply the original version [9] of the ALT or the newest X_Hr(q, 1) version [14] of the ALT to the sequence {MN}.

The reader who is not familiar with the notion of the

‘function’ M) of an n  n normal matrix is referred to [12] and references therein. We shall recall here only the basic definition and property of M). Let

M = ₁P(1) + . . . + _rP(r) (2.8) be the spectral resolution of the normal matrix M, where

₁, . . . , _r are all the distinct eigenvalues of M and P(1), . . . , P(r) are corresponding eigenprojections. Let J be a subset of that contains all the eigenvalues of M and let be a complex-valued function defined on J. Then, we define

(M) by

  (M) = (1)P(1) + . . . + (r)P(r). (2.9) The fact that it is well defined is easily seen by the uniqueness of the spectral resolution.

Let U be an n  n unitary matrix such that

M = U diag(₁, . . . , _n)U^-1 (2.10) where ₁, . . . , _n are all the eigenvalues of M counted with multiplicity, then one gets

  (M) = U diag((₁), . . . ,(_n)) U^-1. (2.11) The Matrix Art and Math Art Programs (using computer

graphic visualization of matrices) in the Fukui project are philosophical and methodical extensions, from science towards art, of Fukui’s approach and also of the Approach via the Aspect of Form and General Topology (cf.

[1,9,12,16] references therein) in the repeat space theory (RST), which is the fundamental unifying theory in the first and second generation Fukui project. In the above-mentioned programs, we often use special n  n matrices, which are referred to as central matrices. The definition of a central matrix is as follows:

Let n  ⁺, let c = (c1, c2)  ². An n  n complex matrix Mis called a central matrix with center c in ² if there exists a p ]0, ] and a function f: [0, [  such that

Mij f i c((  ₁^p j c₂^p) )^1/p (2.12) for all 1  i, j  n, where

1 2 1/

(i c ^ j c ^) ^:= max{i c j c ₁,  ₂}. (2.13) The above central matrix M is denoted by

M = Zen(n, c, p, f). (2.14)

(Remark: The German word ‘Zentrum’ means center.) Let

[0, [

C ^ denote the set of all functions f: [0, [  . Let Mn( ) denote the complex linear space of all n  n complex matrices. One can also define the mapping Zen: ⁺ ² ]0, ]  C^{[0, [} 

n1



Zenn = Zen({n} ² ]0, ]  C^{[0, [} ). (2.16) Let span denote the linear span operation in Mn( ). Then, it is easily seen that

span Zenn = Mn( ). (2.17)

Let n  ⁺. An n  n complex matrix L is called a circle matrix if L is a central matrix with center _n :=

((n + 1)/2, (n + 1)/2). Thus, n  n circle matrix L is expressed in the form:

L = Zen(n, _n, p, f). (2.18) We let On denote the set of all n  n circle matrices. Thus, we have

On = Zen({n}{_n} ]0, ]  C^{[0, [}). (2.19) Matrix

M  span On (2.20)

is called a span circle matrix, or a mandala matrix.

(Mandala is a Sanskrit word that means circle. The basic form of most Hindu and Buddhist mandalas is a square containing circles and squares with a center point.)

An n  n complex matrix M = A1 + A2 + … + Ak, where A1, A2, …, Ak  Zenn, is said to be a k-centered matrix, if M cannot be expressed as a sum of k0 central matrices with k0 < k.

The following d-dimensional generalizations Zen^d and

 ^d of the mapping Zen have applications to quantum chemistry and physics.

Let d  ⁺ with d  2. Define the mapping Zen^d: ⁺ ^d ]0, ]  C^{[0, [} 

1



Zen^d(n, c, p, f) = M(d, n, c, p, f), (2.22) where M = M(d, n, c, p, f) with c = (c1, c2, …, cd)  ^d is an n  n complex matrix given by

M_ij f i c((  ₁^p j c₂ ^p) )^1/p if d = 2;

1 2 3 1/

(( ^{p p} 0 ^p ... 0 ^p) )^p

ij d

M  f i c  j c  c   c if

d  3; (2.23)

for all 1  i, j  n. Here, the right sides of (2.23) with p

=  is given in the manner analogous to (2.13). Note that

Zen = Zen². (2.24)

Let n ⁺ and d  ⁺ with d  2, let

Zennd := Zen^d({n} ^d ]0, ]  C^{[0, [}). (2.25) Then, we have the relationship

Zenn = Zenn2. (2.26)

Let On2 := On, and let

Ond := Zen({n}({_n} ^d-2)  ]0, ]  C^{[0, [}) (2.27) if d  3. An n  n complex matrix Mis called a central matrix centering in ^d if

M Zennd, (2.28)

a circle matrix centering in ^d if

M  Ond, (2.29)

a span circle matrix or a mandala matrix centering in ^d if

M span Ond. (2.30)

Let d  ⁺ with d  2. Let C^Rddenote the set of all functions F: ^d  . Define the mapping

^d: ⁺  ^d  C^{Rd }

1



by

 ^d(n, c, F) = M(d, n, c, F), (2.32) where M = M(d, n, c, F) with c = (c1, c2, …, cd)  ^d is an n

 n complex matrix given by

M_ijF i c j c(  ₁,  ₂) if d = 2;

M_ijF i c j c(  ₁,  ₂,0c₃,...,0c_d)

if d  3; (2.33)

for all 1  i, j  n. Let f  _C^{[0, [}, let p  ]0, ] , and let Fp, f : ^d  denote the function defined by

, ( , ,..., )1 2 (( 1^p 2 ^p ... ^p) ).1/^p

p f d d

F x x x  f x  x   x (2.34)

Then, we have the relationship

^d(n, c, Fp, f ) = Zen^d(n, c, p, f), (2.35) valid for all (n, c, p, f)  ⁺ ^d ]0, ]  C^{[0, [}.

Let n ⁺ and d  ⁺ with d  2, let

_n^d := ^d({n}  ^d  C^Rd). (2.36) An n  n complex matrix M is called the  matrix of index(d, n, c, F,) if

M = ^d(n, c, F). (2.37)

A  matrix of index (d, n, c, F,) is called a vertical matrix if

c1 = c2 = (n + 1)/2, (2.38)

and if

F x x( , ,..., )_{1 2} x_d F x x( ₁, ,..., )₂ x_d (2.39) for all ( , ,..., )x x_{1 2} x_d  ^d.

A  matrix of index (d, n, c, F) is called a horizontal matrix if

c1 = c2 = (n + 1)/2, (2.40)

and if

F x x( , ,..., )_{1 2} x_d F x x( ,₁  ₂,..., )x_d (2.41) for all ( , ,..., )x x_{1 2} x_d  ^d.

The set of all n  n vertical matrices is denoted by Ven and the set of all n  n horizontal matrices is denoted by Hon.

The patterns of the following matrices (i) and (ii) are analyzed in the fundamental part of the Matrix Art Program, in conjunction with the Approach via the Aspect of Form and General Topology in the repeat space theory (RST), which originally stemmed from the experimental and empirical soil of chemistry and engineering: (i) Real-symmetric vertical/horizontal matrices M and (M) with real-valued functions  defined on suitable domains.

(ii) Sum M = A1 + A2 + … + Ak of central matrices A1, A2,

…, Ak, matrix B defined by Bij = (Mij), (M*M), and

((1/2)(M*+M)), with real-valued functions , , 

defined on suitable domains.

The development of the Fukui project along the above line and its implications shall be published elsewhere.

3. AFFIRMATIVE SOLUTION OF PROBLEM 1 In this section, the symbol AC(I) denotes the Banach space of all real-valued absolutely continuous functions on I equipped with the norm given by

|||| = sup {|(t)|: t I} + VI(), where VI() denotes the total variation of  on I, i.e., VI() = sup

i1



(: a = t0 ≤ t1 ≤ . . . ≤ tn = b)

First, we establish theorem 1 by using theorem 2.

Second, we derive theorem 2 from the ALT.

Proof of theorem 1 by using theorem 2. We assume theorem 2. Under the assumption of theorem 2, we show that

(f) = (1/(b - a))(

a

b f()d), (3.1)

and that

(f) = (1/2)(f(b) - f(a)). (3.2) Since f is absolutely continuous it is Riemann integrable, hence dividing both sides of (1.3) by N/(b – a) and letting N

, we see that (3.1) is true.

Now let N: AC(I)  denote the linear functionals defined by

 _N(f) =

r1

N f(x(N, r)) - (1/(b - a))(

a

b f()d))N, (3.3)

N  ^.

Let C¹(I) denote the subspace of AC(I) of all continuously differentiable functions on I. By using Taylor’s theorem, it is not difficult to show that for each f  C¹(I).

  _N(f) (1/2)(f(b) - f(a)) (3.4) as N . (Or, by using the Euler-Maclaurin theorem, C¹ version, one immediately sees that (3.4) is true for all f  C¹(I).)

Define : AC(I)  by

(f) = lim

N _N(f). (3.5)

Note that this functional  is well defined in view of theorem 2. Recall the fact that AC(I) is a Banach space (Cf. [11] and references therein). It is now immediately seen that  is a bounded linear functional by virtue of the

Banach-Steinhouse theorem (the Uniformly Boundedness theorem). Define ^#: AC(I)  by

^#(f) = (1/2)(f(b) - f(a)). (3.6)

Then, it is easily seen that ^# is a bounded linear functional.

Note that

  (f) = ^#(f) (3.7)

for all g  C¹(I). Recall the fact that C¹(I) is a dense subset of AC(I):

C¹(I) = AC(I). (3.8) By the continuity of  and ^#, we see that (f) = ^#(f) for all g  AC(I). Therefore

   = ^#, (3.9)

This completes the proof. //

Proof of theorem 2. By changing the variable, we may assume that a = 0, and b = 1. We have only to verify that if f

 AC[0, 1] then there exist real numbers (f) and (f) such that

1 N

r



1 N



u  g^-1 AC[0, 1]. (3.12)

So, setting f = u  g^-1 in (3.11), we get the conclusion. //

Thus, the goal of the present article has been attained.

We remark that the notion of the normed repeat space reviewed in the appendix unites the approaches via the aspects of form and general topology exploited in a variety of asymptotic analyses of molecular networks in [1-18] and references therein. Equipped with the machinery of Banach algebras and C*-algebras, the notion of normed repeat space with the above-mentioned new unifying power forms a basis of the second generation Fukui project. For a review of the first generation Fukui project, whose basic philosophy we would like to carry on to the second generation project, the reader is referred to ref. [2] entitled

‘Note on the repeat space theory – its development and communications with Prof. Kenichi Fukui’.

4. APPENDIX

Review of the Generalized Repeat Space and the Normed Repeat Space

I. Generalized Repeat Space

There are several equivalent ways of defining the generalized repeat space Xr(q, d) with a given size (q, d) 

+ ⁺. We shall recall below the definition that uses the notion of the sum of subspaces of a linear space (cf. Refs.

[1,13,15,16]).

Fix (q, d)  ⁺ ⁺ and let X (q, d) denote the set of all matrix sequences whose N-th term MNis an arbitrary qN^d qN^d complex matrix, N  ⁺. This set constitutes a -algebra over the field with term-wise addition, scalar multiplication, multiplication

{MN} + {MN} = {MN + MN}, (A.1) k{MN } = {kMN}, (A.2) {MN}{MN} = {MNMN}, (A.3) and involution ()*: X(q, d)X(q, d) defined by

{MN}* = {MN*}, (A.4)

where the * on the right-hand side of (A.4) denotes the adjoint operation.

Let PN denote an N N real-orthogonal matrix given by



P_N  0 1

0 1 0

0 

  0 1

0 0 1

1 0





























.

Let PNa:= (PN -1)^-a where a  {-2, -3, …}. (Note that PNa

equals the transpose of PN-a .)

Let SN denote an N N real idempotent matrix given by



SN 1 0

0 0 0 0

0 0 

  

  0

0 0 0 0

0 0





























. Let PNn denote the N^d N^d matrix given by

PNn = PNn₁ PNn₂ ...  PNn_d, (A.5) where n = (n1, n2, . . . , nd)  ^d, and  denotes the Kronecker product.

Let SNk denote the N^d N^d matrix given by

SNk = SNk₁ SNk₂ ...  SNk_d, (A.6) where k = (k1, k2, ..., kd) ( ⁺ {0})^d.

Let V ^k(q, d) with k = (k1, k2, ... , kd)  {0, 1}^ddenote the subset of X (q, d) defined by

V ^k(q, d) = {{ MN}  X (q, d): m, n  ^d, Q  Mq( ) such that

MN = (P_N^m S_N^k P_Nⁿ) Q for all N >> 0}. (A.7) Let span V ^k(q, d) with k = (k1, k2, ... , kd)  {0, 1}^d denote the linear span of V ^k(q, d).

We defined three fundamental linear subspaces X_r(q, d), X(q, d), and X(q, d) of X (q, d) by X_r(q, d) =

k{0,1} ^dspan V ^k(q, d), (A.8) X_(q, d) = span V ⁰(q, d), (A.9) where 0= (0, 0, ..., 0)  {0, 1}^d, (A.10) X(q, d) =

k{0,1}^d\{0}span V ^k(q, d). (A.11) In (A.8) and (A.11), the  denotes the sum of subspaces in the obvious manner.

We call X_r(q, d), X(q, d), X(q, d), respectively, the generalized repeat space, generalized alpha space, and generalized beta space with size (q, d), and each element of X_r(q, d), X_(q, d), X_(q, d), respectively, a generalized repeat sequence, generalized alpha sequence, and generalized beta sequence with size (q, d).

The following is one of the most fundamental theorems in the repeat space theory.

Theorem A1. For all q, d  ⁺, X_r(q, d) forms a -algebra.

Proof. This was proved in Ref. [15]. //

For the special definition of the generalized repeat space with size (q, 1), set d =1 in the definition of V ^k(q, d) given by (A.7) and observe that

X_(q, 1) = span V ⁰(q, 1)

= span {{MN}  X(q, 1) : m  , Q  Mq( ) such that

MN = PNmQ

for all N >> 0}, (A.12)

X(q, 1) = span V ¹(q, 1)

= span{{MN} X(q, 1): m, n  , Q  Mq( ) such that

MN = (PNmSNPNn) Q

for all N >> 0}, (A.13)

and note that

X_r(q, 1) = X_(q, 1) + X_(q, 1). (A.14)